236 PROCEEDINGS OP THE AMERICAN ACADEMY 



and this represents a cylinder ; because to any radius vector q we can 

 add arwj, where x is any scalar. But what is to become of the asymp- 

 totic cone in this case ? It must also be indeterminate in the same 

 direction, and yet it must still retain tiie property that all radii vectores 

 must lie wholly in its surface. The only surfaces of the second order 

 of which this can be true are pairs of real or imaginary planes. 



The quaternion expression for this is very interesting. If the neg- 

 ative root of a single-sheeted hyperboloid or either root of an ellipsoid 

 vanishes, ihe quadric is represented by the equation 



c.^'^a.T^Q -j- CgS^WoO = 1, 



and becomes an elliptic cylinder. The asymptotic cone 



CjS^aop -\- CgS'Wjp = 0, 

 or 



or 



'^cJ^a.^Q = ± ^ — ^^sSwap, 



becomes a pair of imaginary planes, containing only one real line, 



Q = xa^, 



the line of their intersection, which satisfies the equation because it 

 makes both sides vanish. This may be considered a sort of interme- 

 diate case, on a roundabout road, between the real (hyperbolic) asymp- 

 totic cone and the imaginary (elliptic) one. 



If the positive root of a double-sheeted hyperboloid vanish, the 



surface 



— c^S'^a^o — c.^S'-a.£ = 1 



becomes an imaginary elliptic cylinder. The asymptotic cone 



CjS'«jP — cJ^^u.^Q = 0, 



or 



V — CjSa^Q = ± \c,^8u.^n, 



ao^ain represents a pair of imaginary planes, containing only one real 

 line, 



Q = a:«3, 



this time at right angles to the former direction. 



If, finally, a positive root of a single-sheeted hyperboloid or a nega- 



